Lecture 5 Spectrum Representation DSP First, 2/e Lecture 5 Spectrum Representation

© 2003-2016, JH McClellan & RW Schafer READING ASSIGNMENTS This Lecture: Chapter 3, Section 3-1 Other Reading: Appendix A: Complex Numbers Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer LECTURE OBJECTIVES Sinusoids with DIFFERENT Frequencies SYNTHESIZE by Adding Sinusoids SPECTRUM Representation Graphical Form shows DIFFERENT Freqs Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer FREQUENCY DIAGRAM Want to visualize relationship between frequencies, amplitudes and phases Plot Complex Amplitude vs. Frequency Complex amplitude Spectral line 100 250 –100 –250 f (in Hz) Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer Another FREQ. Diagram A-440 Frequency is the vertical axis Time is the horizontal axis A musical scale consists of a discrete set of frequencies. Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer MOTIVATION Synthesize Complicated Signals Musical Notes Piano uses 3 strings for many notes Chords: play several notes simultaneously Human Speech Vowels have dominant frequencies Application: computer generated speech Can all signals be generated this way? Sum of sinusoids? Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer Fur Elise WAVEFORM Beat Notes Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer Speech Signal: BAT Nearly Periodic in Vowel Region Period is (Approximately) T = 0.0065 sec Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Euler’s Formula Reversed Solve for cosine (or sine) Aug 2016 © 2003-2016, JH McClellan & RW Schafer

INVERSE Euler’s Formula What is the “spectrum” representation for a single sinusoid? Solve Euler’s formula for cosine (or sine) Aug 2016 © 2003-2016, JH McClellan & RW Schafer

SPECTRUM Interpretation Cosine = sum of 2 complex exponentials: One has a positive frequency The other has negative freq. Amplitude of each is half as big Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer GRAPHICAL SPECTRUM w 7 -7 Freq. in rad/s AMPLITUDE, PHASE & FREQUENCY are labels Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer NEGATIVE FREQUENCY Is negative frequency real? Doppler Radar provides intuition Police radar measures speed by using the Doppler shift principle Let’s assume 400Hz 60 mph +400Hz means towards the radar -400Hz means away (opposite direction) Think of a train whistle Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Negative Frequency is still a rotating phasor View as vector rotating counterclockwise q = wt Angle changes vs. time Negative frequency  clockwise rotation Aug 2016 © 2003-2016, JH McClellan & RW Schafer

General form of sinusoid spectrum Amplitudes are multiplied by ½ Complex amplitudes are complex conjugates Called conjugate symmetry Aug 2016 © 2003-2016, JH McClellan & RW Schafer

SPECTRUM Interpretation Cosine = sum of 2 complex exponentials: One has a positive frequency The other has negative freq. Amplitude of each is half as big Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Recall SPECTRUM of cosine w 7 -7 Freq. in rad/s AMPLITUDE, PHASE & FREQUENCY are labels Aug 2016 © 2003-2016, JH McClellan & RW Schafer

REPRESENTATION of SINE Sine = sum of 2 complex exponentials: Positive freq. has phase = -0.5p Negative freq. has phase = +0.5p Aug 2016 © 2003-2016, JH McClellan & RW Schafer

GRAPHICAL Spectrum of sine EXAMPLE of SINE (has Phase of –p/2) w 7 -7 AMPLITUDE, PHASE & FREQUENCY are labels Aug 2016 © 2003-2016, JH McClellan & RW Schafer

SPECTRUM ---> SINUSOID Add the spectrum components: 100 250 –100 –250 f (in Hz) What is the formula for the signal x(t)? Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Gather (A,w,f) information Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz Amplitude & Phase 4 -p/2 7 +p/3 10 0 7 -p/3 4 +p/2 Note the conjugate phase DC is another name for zero-freq component DC component always has f=0 or p (for real x(t) ) Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Add Spectrum Components-1 Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz Amplitude & Phase 4 -p/2 7 +p/3 10 0 7 -p/3 4 +p/2 Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Add Spectrum Components-2 100 250 –100 –250 f (in Hz) Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer Simplify Components Use Euler’s Formula to get REAL sinusoids: Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer FINAL ANSWER So, we get the general form: Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Example: Synthetic Vowel Sum of 5 Frequency Components Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Example: Synthetic Vowel Sum of 5 Frequency Components Aug 2016 © 2003-2016, JH McClellan & RW Schafer

SPECTRUM of VOWEL (Polar Format) 0.5Ak fk Aug 2016 © 2003-2016, JH McClellan & RW Schafer

SPECTRUM of VOWEL (Polar Format) 0.5Ak fk Aug 2016 © 2003-2016, JH McClellan & RW Schafer

Vowel Waveform (sum of all 5 components) One epoch or one period 2 1 -1 -2 Note that the period is 10 ms, which equals 1/f0 Aug 2016 © 2003-2016, JH McClellan & RW Schafer

© 2003-2016, JH McClellan & RW Schafer The VODER by Dudley Resonance controls Stops & plosives Aug 2016 © 2003-2016, JH McClellan & RW Schafer